In most modern formulations of optimal transport, the marginals and optimal coupling are all positive measures.
While other frameworks examine the problem in different ways, positivity remains a necessary feature.
Optimal transport involving signed quantities is comparatively new, but is an active area [1]}, [2]}, [3]}, [4]}, [5]}, [6]}.
For example, applications have arisen in seismic imaging [7]}, [8]}, [9]} and for modelling signed vortices [1]}.
However each application has used an ad hoc approach to make the signed measure into a positive one, e.g. exponentiating the density, adding a large constant to make the density positive, and treating the positive and negative parts separately.
